CN109946029B - A multi-frequency fatigue test method for turbochargers - Google Patents

Abstract

The invention discloses a multi-frequency fatigue test method of a turbocharger. Under the condition of not changing the requirements of the fatigue test, it is different from the commonly used multi-frequency test method of a turbocharger to use the tuning quality, but adopts two or Multiple eigenfrequencies to excite turbocharger turbine rotor blades reduce fatigue test time, reduce problem complexity, and ensure high fidelity of experimental results. In addition, from the perspective of not using the tuning mass, two or more eigenfrequency combinations are used to excite the blades simultaneously, and the spatial damage distribution of the actual turbocharger turbine rotor blades can be simulated more realistically by tuning the excitation force amplitude.

Description

A multi-frequency fatigue test method for turbochargers

technical field

The invention relates to a blade fatigue test method, in particular to a multi-frequency fatigue test method of a turbine rotor blade of a turbocharger, and belongs to the field of testing and testing.

Background technique

The rotor blade is one of the key components of the turbine, which is required to have high reliability, and the service life of the blade is largely determined by its fatigue life. Therefore, fatigue testing of turbine rotor blades is necessary.

The existing turbine blade fatigue test methods are mainly to carry out full-scale blade tests, based on a typical rotating mass to excite the blades to achieve the purpose of resonance excitation, and then realize the fatigue test of the blades. Current testing methods only use the lowest eigenfrequency to induce virtual damage, and additional tuning masses are used to recover the true damage distribution. Even if amplitude amplification techniques are used to perform fatigue tests, it still takes a lot of time. In the existing fatigue testing of turbine rotor blades, the application of tuning mass increases the complexity of fatigue testing, which reduces the natural frequency of the blade, thereby increasing the testing time.

Therefore, from the perspective of reducing fatigue test time and ensuring higher fidelity without using tuning masses, a combination of two or more eigenfrequency combinations is used to excite the blade simultaneously, and the excitation force amplitude can be tuned for more fidelity. A new fatigue test method that simulates the actual spatial damage distribution needs to be established.

SUMMARY OF THE INVENTION

In view of this, the present invention provides a multi-frequency fatigue test method of a turbocharger, which adopts a blade fixing structure, a blade to be tested and an eigenfrequency excitation structure for testing, and one end of the blade to be tested is fixed to the fixed structure above, the eigenfrequency excitation structure is an actuator arranged below the blade to be measured or at least two exciters arranged above the blade to be measured;

Specific steps are as follows:

为基本质量矩阵元素，

为第k个附加调谐质量矩阵，

为第k次调谐元素或激励质量矩阵，EI(z)为摆动弯曲刚度，c(z)为弦长，m(z)为每单位长度的叶片质量，该模型是在多范式数值计算环境Matlabv.15a中生成，它由n＝100个节点组成，L为叶片总长度，在△z＝L/n时等距，△z为相邻节点之间的等距距离，每个节点在x方向上具有一个平移自由度，该模型所示的集中质量系统在双频激励下的动态响应，用点符号表示的非线性二阶微分方程描述如下：Step 1: Discretize the blade to simplify the entire blade problem. The two-dimensional mass model is used to discretize the blade, and the discretized mass model of the blade is obtained, where

is the element of the basic mass matrix,

is the kth additional tuning quality matrix,

is the k-th tuning element or excitation mass matrix, EI(z) is the swing bending stiffness, c(z) is the chord length, m(z) is the blade mass per unit length, the model is in the multi-paradigm numerical computing environment Matlabv .15a, it consists of n=100 nodes, L is the total length of the blade, equidistant when △z=L/n, △z is the equidistant distance between adjacent nodes, each node is in the x direction With one translational degree of freedom on , the dynamic response of the lumped-mass system shown by this model under dual-frequency excitation is described by the nonlinear second-order differential equation represented by dot notation as follows:

为分量速度矢量的绝对值，

为速度矢量，

为加速度矢量；In the formula: M is the total mass matrix, D is the Rayleigh damping coefficient, A is the aerodynamic damping matrix, K is the stiffness matrix, f1, 2 are the nodal excitation force amplitude vectors, ω ₁ and ω ₂ are the first and second types angular natural eigenfrequency of the mode, t is time,

is the absolute value of the component velocity vector,

is the velocity vector,

is the acceleration vector;

Step 2: Solve the stiffness matrix using Euler-Bernoulli theorem. Using first-order Euler-Bernoulli beam theory according to equation (2), semi-analytical processing is performed on each element of the upper triangle of the symmetric compliance matrix (z _i <z _j ), where z is the span coordinate from the base, z _{i, j} are the wingspan coordinates of the i-th and j nodes, using the principle of virtual work, formula (2) represents the deformation of node i caused by the unit load applied by node j, formula (2) by using Simpson's rule Numerical integration is obtained, and the inverse matrix of the compliance matrix can obtain the stiffness matrix shown in Eq. (3),

K=S ^-1 (3)

In the formula, S _ij is the compliance matrix element, S is the compliance matrix,

Step 3:

可以通过如下公式获得：Solve the total mass matrix using the correlation function. Elements of the basic mass matrix of the blade

It can be obtained by the following formula:

式(5)被用于计算相应的分配质量矩阵

where _δij is the Kronecker function, and the lumped-mass model formula requires that additional (tuning or excitation) mass be distributed discretely on its nodes, by distributing the kth mass between its two adjacent nodes i and i+1 Additional quality

Equation (5) is used to calculate the corresponding distribution quality matrix

与

为n维正交基向量，

与

为i维正交基向量的转置矩阵，z_k为第k个翼展坐标，因此，可以获得总质量矩阵M：where z _k is the coordinate of the kth additional mass,

and

is an n-dimensional orthonormal basis vector,

and

is the transpose matrix of the i-dimensional orthonormal basis vector, and z _k is the k-th span coordinate. Therefore, the total mass matrix M can be obtained:

为基本质量矩阵，

为第k个附加调谐或激励质量矩阵，n_t为调谐质量数，In the formula,

is the basic mass matrix,

is the kth additional tuning or excitation mass matrix, n _t is the tuning mass number,

Step 4:

Considering the effect of air damping, the effect of damping is taken into account in the calculation process. Rayleigh damping is expressed in its most common form as follows,

D=aM+bK (7)

In the above formula, a=0 and b=0.0156, a and b are Rayleigh damping coefficients,

The element A _ij of the aerodynamic damping matrix can be obtained by equation (8), where C _d = 2 for the flat plate perpendicular to the flow is taken as a conservative approximation representing the complex aerodynamic situation:

where ρ is the air density at ambient room temperature, c(z) is the chord length, C _d is the aerodynamic drag coefficient at an angle of attack of 90°,

By solving the well-known generalized eigenvalue problem proposed by Eq. (9), the corresponding eigenvectors of the defined model shapes are obtained by Eq. (10), where φ _i is the n-dimensional eigenmode vector of the ith mode: Step 5:

Taking into account the effects of static bending moments ensures a higher reliability of the problem-solving process. To calculate additional parameters, the static bending moment distribution caused by the blade's own weight is calculated through a two-step procedure. First, the shear force Q(z) is calculated using equation (11) when the boundary condition shear force Q(L) = 0 at the tip. :

where g is the gravitational acceleration, m(s) is the blade mass per unit length, then the moment distribution M _S (z) is obtained using equation (12), and the boundary condition M _s (L) = 0:

为第k个调谐质量，

第k个附加质量的翼展坐标，由外部质量自重引起的静态弯矩分布可以通过公式(13)获得，使用叠加原理，可以通过公式(14)得到产生的静态弯矩：where M _S (z) is the swing bending moment caused by self-weight, M _S (L) is the swing bending moment caused by self-weight at length L, Q(s) is the shear force at length s,

is the kth tuning quality,

The wingspan coordinates of the kth additional mass, the static bending moment distribution caused by the self-weight of the external mass can be obtained by formula (13), and using the superposition principle, the generated static bending moment can be obtained by formula (14):

为平均(静态)弯矩，用于损伤计算的节点弯矩历程M(z,t)由公式(15)表示，其中节点曲率通过节点位移历程x(z,t)的二阶导数的中心差分来近似获得：where M _t (z) is the bending moment generated by the tuning mass,

is the average (static) bending moment, the nodal moment history M(z,t) used for damage calculation is expressed by Equation (15), where the nodal curvature is obtained by the central difference of the second derivative of the nodal displacement history x(z,t) to approximate:

Step 6:

在后续讨论的优化过程中被用作约束：Solve for the dual excitation mode mixing degree. Limiting the peak strain amplitude to an empirical sensor type-specific threshold, the rms strain threshold given by Eq. (16)

Used as a constraint in the optimization process discussed later:

where the time span T ₂ -T ₁ corresponds to the steady-state part of the time history and the strain history ε(z,t) of the end fibers of the flange, and the strain history is obtained by a first-order Euler's strain-moment relationship for a Bernoulli beam, as Formula (17) shows:

where x _c (z) is the distance between the end fiber and the bending axis.

In the case of dual frequency excitation, the ratio between the two force amplitudes is measured by the degree of modal mixing, defined here as

In the formula, ψ is the degree of mixing of dual excitation modes, where ψ=1 means pure mode-1 excitation, ψ=0 means pure mode-2 excitation, F ₁ and F ₂ are the excitation force amplitudes,

Step 7:

In the multi-frequency fatigue experiment method, the relationship between the design vector variables and the relevant parameters is established, and the nonlinear constraint of the final fatigue failure is established from the perspective of energy and actual damage distribution.

In the two-frequency method, the design variable vector q consists of two excitation force magnitudes: excitation location and scale factor, which is given by equation (19):

In the formula, w is the scale factor, the total node cycle number vector and the total test time and the total dissipated energy are given by formulas (20) to (24), and the objective function of the multi-frequency method conforms to the formula (25), the multi-frequency case The number of nodal cycles under , varies with the blade, so the nonlinear constraint in Eq. (28) requires that the minimum number of nodal cycles be equal to or greater than the 3 × ¹⁰ cycle basis:

T _tot =w(T ₂ -T ₁ ) (21)

In the formula, N _rs is the number of cycles in the unit rs, n _tot is the total node cycle number vector, T _tot is the total test time, E _tot is the total energy consumption, D is the actual damage index (when D = 1, the fatigue failure is reached. ), E _aer is the aerodynamic energy dissipation, E _str is the structural energy dissipation, D _t is the target damage index

The minimum f(q) is subject to

3×10 ⁶ -n _tot (q)=0 (26)

为时间历程的均方根应变值，x₁₀₀为末端光纤与弯曲轴之间的距离为100mm。In the formula,

is the rms strain value of the time history, and x ₁₀₀ is the distance between the end fiber and the bending axis of 100 mm.

Description of drawings

In order to explain the embodiments of the present invention or the technical solutions in the prior art more clearly, the following briefly introduces the accompanying drawings that need to be used in the description of the embodiments or the prior art. Obviously, the accompanying drawings in the following description are only It is an embodiment of the present invention. For those of ordinary skill in the art, other drawings can also be obtained according to the provided drawings without creative work.

The accompanying drawing of Fig. 1a is a conventional single-frequency fatigue test device;

Figure 1b is a multi-frequency fatigue test device using rotating mass excitation;

Figure 1c shows the ground drive that can be used for single-frequency and multi-frequency excitation;

Figure 2a is a schematic diagram of the plane area of the relevant nodes used for aerodynamic damping;

Figure 2b is a schematic diagram of the associated segment quality;

Figure 2c shows the discretized mass model of the blade.

Among them, 1-fixed structure, 2-blade to be tested, 3-exciter, 4-actuator, 5-plane area of relevant nodes.

Detailed ways

The technical solutions in the embodiments of the present invention will be clearly and completely described below. Obviously, the described embodiments are only a part of the embodiments of the present invention, rather than all the embodiments. All other embodiments obtained by the technical personnel without creative work, all belong to the protection scope of the present invention,

为第k次调谐质量(k＝1,2……)，

为第k个附加质量的翼展坐标，ω_i为第i种模式的角自然特征频率(i＝1,2……)，F_1,2为激振力幅度。图1(a)为工业中最常用的单频疲劳测试装置及方法，基于已有的单频疲劳试验装置，构建新的涡轮装置叶片的多频疲劳试验装置。图1(b)与(c)展示了多频测试方法的两种应用。Figures 1a-1b show different fatigue testing devices and methods for turbine blades, in the figure

is the kth tuning quality (k=1, 2...),

is the wingspan coordinate of the kth additional mass, ω _i is the angular natural characteristic frequency of the i-th mode (i=1, 2...), and F _1,2 is the amplitude of the exciting force. Figure 1(a) shows the most commonly used single-frequency fatigue test device and method in the industry. Based on the existing single-frequency fatigue test device, a new multi-frequency fatigue test device for turbine blades is constructed. Figures 1(b) and (c) show two applications of the multi-frequency test method.

The specific test steps are as follows:

step 1:

为基本质量矩阵元素，

为第k个附加调谐质量矩阵，

为第k次调谐元素或激励质量矩阵，EI(z)为摆动弯曲刚度，c(z)为弦长，m(z)为每单位长度的叶片质量。该模型是在多范式数值计算环境Matlab v.15a中生成的，它由n＝100个节点组成，L为叶片总长度，在△z＝L/n时等距，△z为相邻节点之间的等距距离。每个节点在x方向上具有一个平移自由度。该模型所示的集中质量系统在双频激励下的动态响应，可以用点符号表示的非线性二阶微分方程描述如下：Discretize the blade to simplify the entire blade problem. Based on the existing multi-frequency test device, a two-dimensional mass model is used to discretize the blade, and the discretized quality model of the blade is obtained. As shown in Figure 2, the figure

is the element of the basic mass matrix,

is the kth additional tuning quality matrix,

is the kth tuning element or excitation mass matrix, EI(z) is the oscillating bending stiffness, c(z) is the chord length, and m(z) is the blade mass per unit length. The model is generated in the multi-paradigm numerical computing environment Matlab v.15a, it consists of n=100 nodes, L is the total length of the blade, equidistant when △z=L/n, △z is the distance between adjacent nodes equidistant distance between. Each node has one translational degree of freedom in the x-direction. The dynamic response of the lumped-mass system shown by this model under dual-frequency excitation can be described by a nonlinear second-order differential equation represented by dot notation as follows:

为分量速度矢量的绝对值，

为速度矢量，

为加速度矢量；In the formula: M is the total mass matrix, D is the Rayleigh damping coefficient, A is the aerodynamic damping matrix, K is the stiffness matrix, f _1,2 is the node excitation force amplitude vector, ω ₁ and ω ₂ are the first and second the angular natural eigenfrequencies of the modes, t is time,

is the absolute value of the component velocity vector,

is the velocity vector,

is the acceleration vector;

Step 2:

The stiffness matrix is solved using the Euler-Bernoulli theorem. Using first-order Euler-Bernoulli beam theory according to equation (2), semi-analytical processing is performed on each element of the upper triangle of the symmetric compliance matrix (z _i <z _j ), where z is the span coordinate from the base, z _{i, j} are the wingspan coordinates of the ith and j nodes. Using the principle of virtual work, Equation (2) represents the deformation of node i caused by the unit load applied by node j. Equation (2) is obtained by numerical integration using Simpson's rule, and the inverse matrix of the compliance matrix can obtain the stiffness matrix shown in Equation (3).

K=S ^-1 (3)

In the formula, S _ij is the compliance matrix element, S is the compliance matrix,

Step 3:

可以通过如下公式获得：Solve the total mass matrix using the correlation function. Elements of the basic mass matrix of the blade

It can be obtained by the following formula:

式(5)被用于计算相应的分配质量矩阵，where δ _ij is the Kronecker function, and the lumped-mass model formulation requires that additional (tuned or excited) masses be distributed discretely on its nodes. By distributing the kth additional mass between its two neighbors i and i+1

Equation (5) is used to calculate the corresponding distribution quality matrix,

与

为n维正交基向量，

与

为i维正交基向量的转置矩阵，z_k为第k个翼展坐标，因此，可以获得总质量矩阵M：where z _k is the coordinate of the kth additional mass,

and

is an n-dimensional orthonormal basis vector,

and

is the transpose matrix of the i-dimensional orthonormal basis vector, and z _k is the k-th span coordinate. Therefore, the total mass matrix M can be obtained:

为基本质量矩阵，

为第k个附加调谐或激励质量矩阵，n_t为调谐质量数。In the formula,

is the basic mass matrix,

is the kth additional tuning or excitation mass matrix, and n _t is the tuning mass number.

Step 4:

Considering the effect of air damping, the effect of damping is taken into account in the calculation process. Rayleigh damping is expressed in its most common form as follows,

D=aM+bK (7)

In the above formula, a=0 and b=0.0156, a and b are Rayleigh damping coefficients.

The element A _ij of the aerodynamic damping matrix can be obtained by equation (8), where C _d = 2 for the flat plate perpendicular to the flow is taken as a conservative approximation representing the complex aerodynamic situation:

where ρ is the air density at ambient room temperature, c(z) is the chord length, and C _d is the aerodynamic drag coefficient at an angle of attack of 90°.

By solving the well-known generalized eigenvalue problem proposed by Eq. (9), the corresponding eigenvectors of the defined model shapes are obtained by Eq. (10), where φ _i is the n-dimensional eigenmode vector of the ith mode: Step 5:

Taking into account the effects of static bending moments ensures a higher reliability of the problem-solving process. The calculation of additional parameters is carried out, and the static bending moment distribution caused by the dead weight of the blade is calculated by a two-step procedure.

The shear force Q(z) is first calculated using equation (11) when the boundary condition shear force Q(L) = 0 at the tip:

where g is the gravitational acceleration, m(s) is the blade mass per unit length, then the moment distribution M _S (z) is obtained using equation (12), and the boundary condition M _s (L) = 0:

In the formula, M _S (z) is the swing bending moment caused by self-weight, M _S (L) is the swing bending moment caused by self-weight at length L, Q(s) is the shear force at length s, which is determined by the external The static bending moment distribution caused by the weight of the mass can be obtained by formula (13), and using the superposition principle, the resulting static bending moment can be obtained by formula (14):

为第k个调谐质量，

第k个附加质量的翼展坐标，M_t(z)为由调谐质量产生的弯矩，

为平均(静态)弯矩。用于损伤计算的节点弯矩历程M(z,t)由公式(15)表示，其中节点曲率通过节点位移历程x(z,t)的二阶导数的中心差分来近似获得：In the formula,

is the kth tuning quality,

Wingspan coordinate of the kth additional mass, M _t (z) is the bending moment produced by the tuned mass,

is the average (static) bending moment. The nodal moment history M(z,t) used for damage calculation is represented by Equation (15), where the nodal curvature is approximated by the central difference of the second derivative of the nodal displacement history x(z,t):

Step 6:

在后续讨论的优化过程中被用作约束：Solve for the dual excitation mode mixing degree. The service life of the strain sensor can affect the test time due to test interruptions and equipment replacement. Therefore, the peak strain amplitude is limited to an empirical sensor type-specific threshold, the rms strain threshold given by Eq. (16)

Used as a constraint in the optimization process discussed later:

where the time span T ₂ -T ₁ corresponds to the steady-state part of the time history and the strain history ε(z,t) of the end fibers of the flange, and the strain history is obtained by a first-order Euler's strain-moment relationship for a Bernoulli beam, as Formula (17) shows:

where x _c (z) is the distance between the end fiber and the bending axis.

In the case of dual frequency excitation, the ratio between the two force amplitudes is measured by the degree of modal mixing, defined here as

In the formula, ψ is the degree of mixing of dual excitation modes, where ψ=1 means pure mode-1 excitation, ψ=0 means pure mode-2 excitation, F ₁ and F ₂ are the excitation force amplitudes,

Step 7:

In the multi-frequency fatigue experiment method, the relationship between the design vector variables and the relevant parameters is established, and the nonlinear constraint of the final fatigue failure is established from the perspective of energy and actual damage distribution.

Fatigue testing must satisfy four general constraints, which can be roughly summarized as the minimum number of cycles, the maximum strain threshold, the spatial damage distribution, and the limits of adiabatic heating. To find a solution that satisfies the above constraints, numerical optimization algorithms are required to achieve sufficient accuracy and efficiency. For the current problem, the Matlab optimizer fmincon is used for nonlinear constrained optimization. By default, fmincon computes the Hessian by finite difference, which is reasonable assuming a small finite difference error due to the semi-analytical formulation of the model. In the case of dual frequencies, the blade is excited by two excitation forces simultaneously. In the two-frequency method, the design variable vector q consists of two excitation force magnitudes: excitation location and scale factor, which is given by equation (19):

In the formula, w is the scale factor. The total node cycle number vector and total test time and total dissipated energy are given by equations (20) to (24), and the objective function of the multi-frequency method conforms to equation (25). The number of nodal cycles in the multi-frequency case varies with the blade. Therefore, the nonlinear constraint in Eq. (28) requires that the minimum number of nodal cycles be equal to or greater than the 3×10 ⁶ cycle basis:

T _tot =w(T ₂ -T ₁ ) (21)

In the formula, N _rs is the number of cycles in the unit rs, n _tot is the total node cycle number vector, T _tot is the total test time, E _tot is the total energy consumption, D is the actual damage index (when D = 1, the fatigue failure is reached. ), E _aer is the aerodynamic energy dissipation, E _str is the structural energy dissipation, D _t is the target damage index

The minimum f(q) is subject to

3×10 ⁶ -n _tot (q)=0 (26)

为时间历程的均方根应变值，x₁₀₀为末端光纤与弯曲轴之间的距离为100mm。In the formula,

is the rms strain value of the time history, and x ₁₀₀ is the distance between the end fiber and the bending axis of 100 mm.

The invention discloses a multi-frequency fatigue test method of a turbocharger, which reduces the fatigue test time and ensures high fidelity without using the angle of tuning quality, using two or more eigenfrequency combinations simultaneously Exciting the blade to more realistically simulate the actual spatial damage distribution by tuning the excitation force amplitude

The various embodiments in this specification are described in a progressive manner, and each embodiment focuses on the differences from other embodiments, and the same and similar parts between the various embodiments can be referred to each other. For the devices disclosed in the embodiments In other words, since it corresponds to the method disclosed in the embodiment, the description is relatively simple, and the relevant part can be referred to the description of the method.

The foregoing description of the disclosed embodiments enables any person skilled in the art to make or use the invention, and various modifications to these embodiments will be apparent to those skilled in the art. The principles may be implemented in other embodiments without departing from the spirit or scope of the invention, and thus the invention should not be limited to the embodiments shown herein, but is to be consistent with the principles disclosed herein The widest range consistent with novel features.

Claims (1)

1. A multi-frequency fatigue test method of a turbocharger is characterized in that a blade fixing structure, a blade to be tested and an eigenfrequency excitation structure are adopted for testing, one end of the blade to be tested is fixed on the fixing structure, and the eigenfrequency excitation structure is an actuator arranged below the blade to be tested or at least two vibration exciters arranged above the blade to be tested;

the method comprises the following specific steps:

step 1, discretizing the blade, and discretizing the blade by adopting a two-dimensional cluster mass model to obtain a discretized mass model of the blade, wherein

For the elements of the basic quality matrix,

for the k-th additional tuning quality matrix,

for the k-th tuning element or excited mass matrix, where k is the number of times, i and j are the number of rows and columns, i, j, k is 1,2 … …, ei (z) is the bending stiffness of oscillation, c (z) is the chord length, m (z) is the mass of the blade per unit length, the model is generated in a multi-norm numerical calculation environment Matlab v.15a, and consists of n-100 nodes, L is the total length of the blade, and is equidistant at △ z L/n, △ z is the equidistant distance between adjacent nodes, each node has one degree of translational freedom in the x-direction, and the lumped mass system shown in the model has one degree of translational freedom under dual-frequency excitationThe non-linear second order differential equation represented by a dot symbol is described as follows:

wherein M is a total mass matrix, D is a Rayleigh damping coefficient, A is a pneumatic damping matrix, K is a stiffness matrix, f₁And f₂Is the first and second node excitation force amplitude vector, omega₁And omega₂The angular eigenfrequencies of the first and second modes, x is the displacement vector, t is the time,

is the absolute value of the component velocity vector, wherein,

is the velocity vector of the first component,

is the absolute value velocity vector of the first component,

is the velocity vector of the nth component,

is the absolute value velocity vector of the nth component,

meaning a velocity vector of 2,3 … … components,

in the form of a velocity vector, the velocity vector,

is an acceleration vector;

step 2: solving the stiffness matrix by using the Euler-Bernoulli theorem, and performing semi-analytic processing on each element of an upper triangle of the symmetrical flexibility matrix by using a first-order Euler-Bernoulli beam theory according to an equation (2), wherein z is a wingspan coordinate from the base, and z is a wingspan coordinate_iAnd z and_jspanwise coordinates for the ith and j nodes, z_i﹤z_jUsing the principle of imaginary work, equation (2) represents the deformation of the node i caused by the unit load applied by the node j, equation (2) is obtained by numerical integration using the Simpson's rule, and the inverse matrix of the compliance matrix can give the stiffness matrix shown in equation ⑶,

K＝S^-1(3)

in the formula, S_ijIs a compliance matrix element, S is a compliance matrix,

and step 3:

solving the total mass matrix, basic mass matrix elements, using a correlation function

Can be obtained by the following formula:

in the formula (I), the compound is shown in the specification,_ijfor the Kronecker function, the lumped mass model formula requires that the tuning or excitation mass be distributed discretely over its nodes by distributing the kth tuning mass between its two neighboring nodes i and i +1

Equation (5) is used to calculate the corresponding distribution quality matrix

In the formula, z_kFor the coordinates of the kth additional mass,

and

is an n-dimensional orthogonal basis vector,

and

is a transposed matrix of the i-dimensional orthogonal basis vectors,

for the kth spanwise coordinate, the total mass matrix M can thus be obtained:

in the formula (I), the compound is shown in the specification,

in order to be a basic quality matrix,

for the kth additional tuning or excitation quality matrix, n_tIn order to tune the mass number,

and 4, step 4:

the influence of air damping, which is taken into account in the calculation, is expressed in its most common form as rayleigh damping, as shown in detail below,

D＝aM+bK (7)

in the above formula, a is 0 and b is 0.0156, a and b are rayleigh damping coefficients,

element A of the aerodynamic damping matrix_ijCan be obtained by the formula (8) wherein C is perpendicular to the flow of the flat plate_dAs a conservative approximation representing a complex aerodynamic situation:

where ρ is the air density at ambient room temperature, C (z) is the chord length, C_dIs an aerodynamic drag coefficient of an angle of attack of 90 degrees,

by solving the generalized eigenvalue problem set forth in equation (9), the corresponding eigenvectors of the defined model shape are derived from equation (10), where φ_iN-dimensional eigenmode vectors for the ith mode:

and 5:

considering the influence of the static bending moment, ensuring higher reliability of the solving process, calculating additional parameters, calculating the static bending moment distribution caused by the self weight of the blade by a two-step procedure, and firstly calculating the shearing force Q (z) by using the equation (11) when the boundary condition shearing force Q (L) of the tip is 0:

where g is the gravitational acceleration, m(s) is the blade mass per unit length, then the moment distribution Ms (z) is obtained using equation (12), and the boundary condition Ms (L) is 0:

where Ms (z) is the bending moment of sway due to self weight, Ms (L) is the bending moment of sway due to self weight at length L, q(s) is the shear force at length s, the static bending moment distribution due to self weight of the external mass can be obtained by equation (13), and the resulting static bending moment can be obtained by equation (14) using the principle of superposition:

in the formula (I), the compound is shown in the specification,

for the k-th tuning mass,

spanwise coordinate of the kth additional mass, M_t(z) is the bending moment created by the tuned mass,

is static bending moment, n_iIs the maximum value of k, indicating the kth tuning quality

Has a minimum value of 1 and a maximum value of n_iThe node bending moment history M (z, t) used for damage calculation is represented by equation (15), where the node curvature is approximated by the central difference of the second derivative of the node displacement history x (z, t):

step 6:

solving the degree of mixing of the double excitation modes, and limiting the peak strain amplitude to experienceSensor type specific threshold value, root mean square strain threshold value given by equation (16)

Used as constraints in the optimization process discussed later:

wherein the time span T₂-T₁The strain history (z, t) of the end fiber corresponding to the steady-state portion of the time history and the flange, which yields bernoulli beams through the strain-moment relationship of first order Euler, as shown in equation (17):

in the formula, x_c(z) is the distance between the terminal optical fiber and the bending axis;

in the case of dual-frequency excitation, the ratio between the two force amplitudes is measured by the modal mixing degree, defined here as

Where ψ is a degree of mixing of the two excitation modes, where ψ ═ 1 denotes F since there are two excitations₁Far greater than F₂Time-domain excitation, also known as pure mode-1 excitation; psi-0 denotes F₂Excitation at 0, also called pure mode-2 excitation, F₁And F₂In order to obtain the amplitude of the exciting force,

and 7:

in the multi-frequency fatigue test method, a relational expression of design vector variables and related parameters is established, and the nonlinear constraint of final fatigue failure is established from the angle of energy and actual damage distribution;

in the dual-frequency method, the design variable vector q consists of two excitation forces: excitation position and scale factor, given by equation (19):

where w is a scale factor, the vector of the total node cycle number and the total test time and total dissipated energy are given by equations (20) to (24), and the objective function of the multi-frequency method conforms to equation (25), the node cycle number in the case of multi-frequency varies with the blade variation, and therefore, the non-linear constraint in equation (28) requires that the minimum node cycle number be equal to or greater than 3X10⁶And (3) cyclic reference:

T_tot＝w(T₂-T₁) (21)

in the formula, N_rsIs the number of cycles in unit rs, n_totAs vector of total node cycle number, T_totFor total test time, E_totFor total energy consumption, D is the actual injury index, E_aerFor aerodynamic energy dissipation, E_strFor structural energy dissipation, D_tFor the target damage index, f (q) is the algorithm optimization coefficient when the design variable vector is q, D_t(z_i) Is a spanwise coordinate of z_iTarget injury index, D (z)_i) Is a spanwise coordinate of z_iReal injury fingerThe number of the first and second groups is,

minimum f (q) obedience

3×10⁶-n_tot(q)＝0 (26)

In the formula (I), the compound is shown in the specification,

root mean square strain value, x, of time history₁₀₀(q) is the 100 th displacement vector in the design of the variable vector q, x₁₀₀The distance between the end fiber and the bending axis was 100 mm.

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